Experienced credit risk professionals with a lot of modeling background soon come to appreciate that there is an important distinction between the model that predicts the most accurate default probability term structure for a company now and the model that best predicts that term structure at some point in the future. There are a number of key points that highlight the distinction:
1. The best default probability term structure now can be predicted with full knowledge of inputs like the firm’s financial ratios and stock price history, in addition to the macro factor history.
2. When generating a default term structure for the same company at a date 10 years from now, the firm’s financial ratios and stock price history will be unknown.
3. There are two ways to deal with this issue. The first way, to econometrically predict the unknown inputs, has been labeled a “forbidden model” by Angrist and Pischke, who wrote that this approach produces biased coefficients and has been banned in the Economics Department at MIT since 1975. See Angrist and Pischke  for details. The use of forecasted inputs that predict the value of unobservable company assets in the Merton model framework is a widely used but classic example of a “forbidden model.”
4. The second approach is to use a ‘reduced reduced form” model that uses as inputs only those variables that are assumed to be known at the relevant point in time. We give the examples for Exxon Mobil (XOM) and Daimondback Energy, Inc. (FANG) in this brief note.
5. Like all regression analysis, our prediction of the default term structure conditional on the values of macro factors describes the correlation between macro factors and default probabilities, NOT CAUSATION.
6. To accurately model this correlation among factors and default probabilities, most econometricians we work with would not drop a variable that couldn’t possibly be a “cause” of default probability movement, because we are not predicting the “cause” of the link between oil price movements, other macro factors, and default probabilities: the cause of oil price movements is unknowable and involves the extremely complex interaction of supply and demand, which in turn affects the firms we care about.
7. The mathematical equation used to model this correlation is commonly selected to be the same logistic distribution or the similar normal distribution, two popular “link functions” in generalized linear models. Both distributions produce similar results. See Gelman et al  for details.
8. In the graphic above, we show the list of macro factors that have had persistent correlation with the default probabilities of Exxon Mobil over 268 months for which we have default probability data.
9. Oil prices are statistically significant, and, as we could expect, a rise in oil prices causes the default probabilities to decline.
For readers who want more technical details on how this calculation is done in the Kamakura Risk Information Services default probability and bond information service, we call the reader’s attention to examples published previously:
van Deventer, Donald R. "Bank of America and CCAR Stress-Testing: A Simple Worked Example,” Kamakura Corporation Blog, February 11, 2016.
Van Deventer, Donald R. "CCAR Stress Tests for 2016: A Wells Fargo & Co. Example of Effective Challenge for Default Probability Stress-Testing,” Kamakura Corporation Blog, April 25, 2016.
The graph below shows the results for DiamondBack Energy, Inc. (FANG):
In this case, we have only 42 months of history. This is the reason why oil prices have not shown up as statistically significant in this regression. In subsequent installments in this series, we talk about other issues that arise in linking macro factor movements to default probabilities. We close with one of the standard choices for a macro factor universe that is a user-defined option in KRIS, a five-region set of macro economic factors and analytical functions of them that have been very popular among KRIS clients as a compact set of factors with high explanatory power. The candidate variables for the two regressions above were selected from this list:
Using the econometric relationships already in KRIS and the Kamakura Risk Manager Version 10-based simulation capabilities in KRIS Credit Portfolio Manager, clients can simulate the exposure of a fixed income portfolio to changes in macro factors and their impact on probabilities of default.
Joshua D. Angrist and Jorn-Steffen Pischke, Mostly Harmless Econometrics: An Empiricist’s Companion, Princeton University Press, Princeton, 2009.
Sreedhar T. Bharath and Tyler Shumway, “Forecasting Default with the Merton Distance to Default Model,” Review of Financial Studies, May 2008. Original paper dated 2004.
J. Y. Campbell, Jens Hilscher, and Jan Szilagyi, “In Search of Distress Risk,” Journal of Finance, December 2008. Original working paper was dated October 2004.
Gregory Duffee, “Estimating the Price of Default Risk,” The Review of Financial Studies, 12 (1), 197-226, Spring 1999.
D. Duffie, M. Schroder and C. Skiadas, “Recursive Valuation of Defaultable Securities and the Timing of Resolution of Uncertainty,” working paper, Stanford University, 1995.
D. Duffie and K. Singleton, “Modeling Term Structures of Defaultable Bonds,” Review of Financial Studies, 12(4), 1999, 197-226.
Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald R. Rubin, Bayesian Data Analysis, Third Edition, Chapman & Hall/CRC Texts in Statistical Science, 2013.
J. Hilscher, Robert A. Jarrow, and Donald R. van Deventer, “Measuring the Risk of Default, A Modern Approach“, RMA Journal, July-August, 2008, pp. 60-65.
J. Hilscher and M. Wilson, “Credit Ratings and Credit Risk: Is One Measure Enough?” Working Paper, Brandeis University and Oxford University, 2013.
J. Ingersoll, Theory of Modern Financial Decision Making (Rowman & Littlefield Publishers, Inc., New York), 1987.
R. Jarrow, “Default Parameter Estimation using Market Prices,” Financial Analysts Journal, 2001.
R. Jarrow, D. Lando and S. Turnbull, “A Markov Model for the Term Structure of Credit Risk Spreads,” The Review of Financial Studies, 10(2), 1997, 481-523.
R. Jarrow, David Lando, and Fan Yu, “Default Risk and Diversification: Theory and Applications,” Mathematical Finance, January 2005, 1-26
R. Jarrow and S. Turnbull, “Pricing Derivatives on Financial Securities Subject to Credit Risk,” Journal of Finance, 50(1), 1995, 53 – 85.
R. Jarrow and S. Turnbull, Derivative Securities, 2nd edition, South-Western College Publishing: Cincinnati, Ohio, 2000.
R. Jarrow, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 2.0, May, 2003.
R. Jarrow, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 3.0, December, 2003.
R. Jarrow, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 4.1, Edition 10.0, January 26, 2006.
R. Jarrow, S.P. Klein, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 5.0, Edition 12.0, March 3, 2011.
R. Jarrow, J. Hilscher, T. Le, M. Mesler and D. van Deventer, Kamakura Public Firm Default Probabilities Technical Guide, Kamakura Risk Information Services, Version 6.0, Edition 7.0, June 30, 2015.
R. Jarrow and Donald R. van Deventer, “Integrating Interest Rate Risk and Credit Risk in ALM,” Asset & Liability Management: A Synthesis of New Methodologies, RISK Publications, 1998.
R. Jarrow and Donald R. van Deventer, "Practical Use of Credit Risk Models in Loan Portfolio and Counterparty Exposure Management: An Update," Credit Risk: Models and Management, Second Edition, David Shimko, Editor, RISK Publications, 2004.
R. Jarrow, Donald R. van Deventer and Xiaoming Wang, “A Robust Test of Merton’s Structural Model for Credit Risk,” Journal of Risk, Fall 2003, pp. 39-58.
J. Johnston, Econometric Methods, 2nd edition, McGraw-Hill Book Company, 1972.
E.P. Jones, S. P. Mason, and E. Rosenfeld, “Contingent Claims Analysis of Corporate Capital Structure: An Empirical Investigation,” Journal of Finance 39, 611-626, 1984.
F. A. Longstaff and E. S. Schwartz, “A Simple Approach to Valuing Risky Fixed and Floating Rate Debt”, Journal of Finance 50, 789-820, 1995.
G. S. Maddala, Introduction to Econometrics, 2nd edition, Prentice Hall, 1992.
L. Nielsen, J. Saa-Requejo and P. Santa-Clara, “Default Risk and Interest Rate Risk: The Term Structure of Default Spreads,” working paper INSEAD, 1993.
van Deventer, Donald R., K. Imai and M. Mesler, Advanced Financial Risk Management: Tools and Techniques for Integrated Credit Risk and Interest Rate Risk Management., second edition, John Wiley & Sons, Singapore, 2013.
van Deventer, Donald R.” ‘Point in Time’ versus ‘Through the Cycle’ Credit Ratings: A Distinction without a Difference,” Kamakura blog, www.kamakuraco.com, March 24, 2009.